- #1

Luqman Saleem

- 18

- 3

[tex]H=-t\sum_{<i,j>}(c_i^\dagger c_j+h.c.)+U\sum (n_i n_j)[/tex]

I solved this Hamiltonian using MATLAB. It was quite simple. t and U are hopping and interaction potentials. c, [tex]c^\dagger [/tex] and n are annihilation, creation and number operators in real space respectively. The first term is hopping and 2nd is two-body interaction term. <i,j> is indicating that hopping is possible only to nearest neighbors. To solve this Hamiltonian I break it down as: (for M=# for sites=2 and N=# of particles=1)

[tex]H=-t (c_1^\dagger c_2 + c_2^\dagger c_1)+U n_1 n_2 [/tex]

The basis vectors that can be written in binary notation are:

01, 10

Using t=1, U=1 and above basis the Hamiltonian can be written as:

H=[0 -1

-1 0]

That is correct.

I checked with different values of M,N,U and t this MATLAB program give correct results.

[tex]**In K-space**[/tex]

To diagonalize this Hamiltonian in K-space we can perform Fourier transform of operators that will results in:

[tex]H(k)=\sum_k \epsilon_k n_k + U / L \sum_ {k,k,q} c_k^\dagger c_{k-q} c_{k'}^\dagger c_{k'+q}[/tex]

Where [tex]\epsilon_k=-2tcos(k)[/tex].

To diagonalize this Hamiltonian I make basis by taking k-points between -pi and +pi (first brillion zone) i.e. for M=2 and N=1 allowed k-points are: [0,pi]

Here first term is simple to solve and I have solved it already but I can't solve the 2nd term as it includes summation over three variables.

To get in more details of my attempt you can see https://physics.stackexchange.com/q/352833/140141

[tex]**My question:**[/tex]

1. What is physical significance of 2nd term in H(k) given above? I mean what is it telling about which particles are hopping from where to where? What are limits on q, k and k'?

2. If you think any article can help me with this problem then please tell me about that.Thanks a lot.